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I'm currently taking a class about model building in applied math. We spent quite a bit of time on Buckingham $\Pi$ theorem. I understand it on a shallow level. I know how to solve the homework and quiz problems but I just don't understand why it works.

My next big assignment is a presentation and essay. Buckingham's theorem is a big part of it. More specifically, we are given a problem with the following variables:

\begin{array} \hline D & Diameter \\ \hline V & Velocity \\ \hline \rho & Density \\ \hline F & Force \\ \hline n & Revolutions/second \\ \hline g & Acceleration\phantom{-}due\phantom{-}to\phantom{-}gravity \\ \hline \mu & Viscosity \\ \hline \end{array}

We are asked to use Buckingham's theorem to derive the following equation:

$$ F = \rho D^2 v^2 \phi(\frac{nD}{V},\frac{gD}{V^2}, \frac{\mu}{\rho D V})$$

where $\phi$ is a function. I know how to get the answer but I can't tell you why it works. This was fine when it was just homework or quizzes problems but now I have to do a presentation on it. Why does Buckingham's theorem work here? From what I understand it doesn't always work - sometimes it gives you a formula that has no coherent physical meaning. We have to use theory from physics to justify the use of Buckingham's theorem but what would this justification look like? This problem is about a ship and the force of its propeller if that makes a difference. Furthermore, what exactly is the relationship between the equation above and the actual equation from the Buckingham theorem $f(\Pi_1, \Pi_2, \Pi_3, \Pi_4) = 0$?

Bain
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  • Ceterum censeo: Intuition can't be explained by definition. It can be acquired by practice (i.e. by doing mathematics), but not by asking at MSE. Questions containing the word "intuition" should be automatically deleted. –  Nov 04 '20 at 16:36
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    @ProfessorVector I completely disagree. Intuitive explanations are far from the whole story, but they're not worthless. Questions about intuition are some of the most valuable, and far more interesting to me than yet another "do this calculation for me" type. A given explanation may or may not work for a given audience, but they are crucial for understanding. – Jair Taylor Nov 04 '20 at 17:25
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    @Professor Vector I don't understand the target of your remark. If 'intuition' is an allowed tag, why should questions invoking 'intuition' be banned ? I highly praise myself all the elements that give me more intuition in the general sense of 'insight', connection with a different sector of mathematics, etc. – Jean Marie Nov 04 '20 at 17:26
  • Besides, are we at the eve of a punic war ? :) – Jean Marie Nov 04 '20 at 17:29
  • @Jean Marie Concerning "intuition can't be explained", just look up "intuition" in the dictionary of your trust. If it can't be summed up with "irrational", give me a link, please. What is allowed here is one thing, my opinion is a different thing. Thus, questions like this can provoke opinionated responses. That could be a reason to close a question, but... consistency is the excuse of a small mind. =) –  Nov 04 '20 at 18:35
  • I don't want to polemicate. The cause of misunderstanding may come from the fact that French (my mother language) and English do not attribute the same nuances to a word which is the same in the two languages, but for me 'intuition' is not orthogonal to 'rationality'... – Jean Marie Nov 04 '20 at 18:41
  • @ProfessorVector okay then can you address literally any of the questions I asked that didn't use this word? I can repost it without the word but this seems pointless. – Bain Nov 04 '20 at 18:51
  • It is pointless, because "I know how to get the answer but I can't tell you why it works" is just the same thing, no matter how you formulate it. You can't. You acquire intuition (not accidentally the contrary of "tuition") by doing maths, not by watching it done, not by being explained at MSE. My first (or second) language isn't English, either, but it's the common language, here, and it defines "intuition" as "the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference" https://www.merriam-webster.com/dictionary/intuition Good luck. –  Nov 04 '20 at 19:55
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    Good grief, I don't think "proof by dictionary" is one I was aware of. You may have your opinions about intuition, but why do you need to drag others down and stop them from trying to help one another? I've gotten tons of insight, or intuition, whatever you want to call it, from seeing other people explain themselves. You don't have to police everybody, just let people discuss and grow. – mpettis Nov 25 '20 at 19:49

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Using Buckingham $\Pi$ theorem is like working a mechanical puzzle: enter image description here (Source: https://en.wikipedia.org/wiki/Mechanical_puzzle)

You employ all your backgrounds (intuition + ...) to solve it.

Victor Quezada
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